Question

1. the formula for the area of a trapezoid is ( a=\frac{1}{2} hleft(b_{1}+b_{2}

ight) ). express ( b_{1} ) in terms of ( a, h ) and ( b_{2} ).

1. the equation for the volume of a cylinder is ( v=pi r^{2} h ). the positive value of ( r ), in terms of ( h ) and ( v ), is

(1) ( r=sqrt{\frac{v}{pi h}} ) (2) ( r=sqrt{v pi h} ) (3) ( r = 2 v pi h ) (4) ( r=\frac{v}{2 pi} )

1. the formula for electrical power, ( p ), is ( p = i^{2} r ), where ( i ) is current and ( r ) is resistance. the formula for ( i ) in terms of ( p ) and ( r ) is

(1) ( i=left(\frac{p}{r}
ight)^{2} ) (3) ( i=(p - r)^{3} )
(2) ( i=sqrt{\frac{p}{r}} ) (4) ( i=sqrt{p - r} )

1. the formula for the volume of a cone is ( v=\frac{1}{3} pi r^{2} h ). solve the equation for ( h ) in terms of ( v, r ) and ( pi ).

Explanation:

Step1: Solve for \(b_1\) in \(A=\frac{1}{2}h(b_1 + b_2)\)

Multiply both sides by \(2\): \(2A=h(b_1 + b_2)\)

Step2: Divide both sides by \(h\)

\(\frac{2A}{h}=b_1 + b_2\)

Step3: Subtract \(b_2\) from both sides

\(b_1=\frac{2A}{h}-b_2\)

Step4: Solve \(V = \pi r^{2}h\) for \(r\) (positive value)

Divide both sides by \(\pi h\): \(\frac{V}{\pi h}=r^{2}\)
Take the square - root of both sides: \(r=\sqrt{\frac{V}{\pi h}}\) (matches option (1))

Step5: Solve \(P = I^{2}R\) for \(I\)

Divide both sides by \(R\): \(\frac{P}{R}=I^{2}\)
Take the square - root of both sides: \(I=\sqrt{\frac{P}{R}}\) (matches option (2))

Step6: Solve \(V=\frac{1}{3}\pi r^{2}h\) for \(h\)

Multiply both sides by \(3\): \(3V=\pi r^{2}h\)
Divide both sides by \(\pi r^{2}\): \(h = \frac{3V}{\pi r^{2}}\)

Answer:

1. \(b_1=\frac{2A}{h}-b_2\)
2. \(r=\sqrt{\frac{V}{\pi h}}\) (Option (1))
3. \(I=\sqrt{\frac{P}{R}}\) (Option (2))
4. \(h=\frac{3V}{\pi r^{2}}\)